Categories for the Working Mathematician says
Given two functors $S, T: C \to B$, a natural transformation $r: S \to\to T$ is a function which assigns to each object $c$ of $C$ an arrow $\tau_c = \tau c : Sc\to Tc$ of $B$ in such a way that every arrow $f: c\to c'$ in $C$ yields a diagram
Given any two functors $S,T: C\to B$, is there always a natural transformation $r: S \to\to T$ ? Specifically, for each $c \in C$, is there always an arrow $Sc\to Tc$?
When there is, is such a natural transformation unique? Specifically, for each $c \in C$, is such an arrow $Sc\to Tc$ unique?
Thanks.
Let $I = 0\to 1$ be the category with two elements and a single non-identity arrow. Let $S: I \to I$ take both elements to 1, and let $T: I \to I$ take both elements to 0. Then there is no natural transformation $S\Rightarrow T$.
The answer to uniqueness is also no, I'll leave that as an exercise but I'd be happy to provide an example if you'd like.